The singularities for a periodic transport equation

In this paper, we consider a 1D periodic transport equation with nonlocal flux and fractional dissipation $$u_{t}-(Hu)_{x}u_{x}+\kappa\Lambda^{\alpha}u=0,\quad (t,x)\in R^{+}\times S,$$ where $\kappa\geq0$, $0<\alpha\leq1$ and $S=[-\pi,\pi]$. We first establish the local-in-time well-posedness for this transport equation in $H^{3}(S)$. In the case of $\kappa=0$, we deduce that the solution, starting from the smooth and odd initial data, will develop into singularity in finite time. If adding a weak dissipation term $\kappa\Lambda^{\alpha}u$, we also prove that the finite time blowup would occur.